# Charged dielectric sheet between two grounded conductors

1. Aug 31, 2011

### Termotanque

I have a proposed solution to the following problem, but I don't know if is correct, and so I'll ask you for help:

A thin dielectric sheet of uniform charge density $\sigma$ (I'll assume $\sigma > 0$ for simplicity) is surrounded by two thin conductors, which are initially uncharged and are connected by a wire. The first and second conductors are at distances $a, b$ respectively from the dielectric. The three sheets are parallel. The configuration is then like a sandwich. I want to find the electric field in every point of the space.

My proposed solution is that the electric field between the conductor A and the dielectric, and between the conductor B and the dielectric, is null. The electric field is $|\bar{E}_{outside}| = \dfrac{\sigma}{2\epsilon_0}$ everywhere else.

My argumentation is as follows: for the electric field outside the sandwich, the solution is clear and comes from Gauss's law.

For the electric field inside, let's suppose that a charge $\eta$ is induced on the sheet A (and thus a charge $-\eta$ on B). This means that the electric field in the space between A and the dielectric will be $E_{A} = \dfrac{\sigma+\eta}{2\epsilon_0}$, while $E_{B} = \dfrac{\sigma-\eta}{2\epsilon_0}$. The potential difference between A and B is then $\Delta V = E_{B} b - E_{A} a$.

But the hypothesis is that $\Delta V = 0$ (they are connected with a wire), and so the difference above only holds is $\eta = \sigma$, which means that $E_{A} = E_{B} = 0$.